Solutions of second order multi-point boundary value problems

We consider classes of second order boundary value problems with a nonlinearity f (t, x) in the equations and subject to a multi-point boundary condition. Criteria are established for the existence of nontrivial solutions, positive solutions, and negative solutions of the problems under consideration. The symmetry of solutions is also studied. Conditions are determined by the relationship between the behavior of the quotient f(t, x)/x for x near 0 and infinity, and the largest positive eigenvalue of a related linear integral operator. Our analysis mainly relies on the topological degree and fixed point index theories.